The decomposition of (REF ) into the factors \(C(n,t,s)\)  and \(R(n,t,s)\)  is useful to understand the role of the prior distribution over \(\alpha \) , and to compare our results with the one of [1]}, [2]}. In particular, the term \(R(n,t,s)\)  does not depend on \(\alpha \)  and, hence, on the choice of \(\pi \) . This is indeed the key term studied in [2]}, where it is shown that, under some assumptions, \(\liminf R(n,t,s)>0\)  as \(n\rightarrow \infty \)  in \(P^{(\infty )}\) -probability, for \(t < s\) . On the contrary, \(C(n,t,s)\)  incorporates information about \(\alpha \)  and its prior distribution.
In the fixed \(\alpha \)  case, which can be thought of as having a degenerate prior \(\pi =\delta _{\alpha }\)  for some \(\alpha >0\) , the term \(C(n,t,s)\)  boils down to \(\alpha ^{s-t}\)  which is constant with respect to \(n\) . This is sufficient for [2]} to deduce lack of consistency for fixed \(\alpha \) , which means that
\(\lim \sup \, \text{pr}( K_n = t \mid X_{1:n},\alpha ) <1\) 
